01.04.1
Chapter 01.04
Binary Representation of Numbers
After reading this chapter, you should be able to:
1.
convert a base-10 real number to its binary representation,
2.
convert a binary number to an equivalent base-10 number.
In everyday life, we use a number system with a base of 10.
For example, look at the
number 257.56.
Each digit in 257.56 has a value of 0 through 9 and has a place value.
It can
be written as
2
1
0
1
2
10
6
10
7
10
7
10
5
10
2
76
.
257
−
−
×
+
×
+
×
+
×
+
×
=
In a binary system, we have a similar system where the base is made of only two digits 0 and
1. So it is a base 2 system.
A number like (1011.0011) in base-2 represents the decimal
number as
( )
1875
.
11
)
2
1
2
1
2
0
2
0
(
)
2
1
2
1
2
0
2
1
(
)
0011
.
1011
(
10
4
3
2
1
0
1
2
3
2
=
×
+
×
+
×
+
×
+
×
+
×
+
×
+
×
=
−
−
−
−
in the decimal system.
To understand the binary system, we need to be able to convert binary numbers to
decimal numbers and vice-versa.
We have already seen an example of how binary numbers are converted to decimal
numbers. Let us see how we can convert a decimal number to a binary number. For example
take the decimal number 11.1875.
First, look at the integer part: 11.
1.
Divide 11 by 2.
This gives a quotient of 5 and a remainder of 1.
Since the
remainder is 1,
1
0
=
a
.
2.
Divide the quotient 5 by 2.
This gives a quotient of 2 and a remainder of 1.
Since
the remainder is 1,
1
1
=
a
.
3.
Divide the quotient 2 by 2.
This gives a quotient of 1 and a remainder of 0.
Since
the remainder is 0,
0
2
=
a
.
4.
Divide the quotient 1 by 2.
This gives a quotient of 0 and a remainder of 1.
Since
the remainder is ,
1
3
=
a
.
Since the quotient now is 0, the process is stopped.
The above steps are summarized in Table
1.

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